Calculus 12 - class review

Today, Pari and I looked at the curriculum for calculus 12 (taught in high school).  We looked at the content material and the way in which the curriculum was structured.

First of all, the content was similar to the one Peri and I had studied in our high school calculus classes.  The main focus of the material is on differentiation and integration.  However, there were some novel features.  The material included a section dealing with the historical foundation of calculus.  Students were expected to know about the different mathematicians who contributed substantially to the material presented.  Newton, DesCartes, Leibniz, and others, are studied.  This brings history into the context of mathematics.

Another feature of the curriculum was its emphasis on assessment of the material taught.  The curriculum provides criteria by which the students can self-assess their own or others' work.  The assessment criteria are specific to each section of the curriculum.  Students are expected to work out problems on their own or with others, and to develop innovative ways to both answer problems and to self-correct their own problem-solving skills.

Micro-teaching reflection - Greek Alphabet

I really enjoyed doing this micro-teaching.  I wanted to communicate to students a bit of my knowledge of Greek and particularly of the Greek alphabet.  Math students see Greek letters all the time in their classes, but usually do not get to learn about what these letters mean exactly.  As far as the lesson was concerned, the main thing I noticed was how quickly time went by.  I did not have the time to do everything that I wanted to do.  In fact, I didn't even come close.  The activity which I had planned to give students, which would involve them writing out the Greek letters on paper, did not materialize.  I found that the most efficient thing was just to write all the letters on the board.  Thus, in order to save time, I had to make the lesson less interactive.  If I were to do it again, I would try to focus on less content, and make it more interactive by providing enough time for an activity.  Most of the comments I received from my peers had to do we these two things: time management, and lack of interactive activity.  However, I did manage during my microteaching to ask lots of questions, which is usually what I do when I teach.  I love asking questions and probing students for answers.  I like using the blackboard a lot, and I think that for this micro-teaching, this proved beneficial.  I also trying to make the lesson interesting by giving a lot of practical knowledge to students - how to pronounce certain words and letters in Greek, and informing them that their former training was often incorrect (math teachers often don't know how to pronounce Greek!).  This made it possible for students to connect with my lesson and to draw applications for the future.

Battleground Schools - Entrance Slip

I found the history of the approaches to teaching mathematics to be very interesting.  I grew up in the 90s, which, according to the article, was a period of time when the NCTM Standards' approach to mathematics was prevalent.  I grew up with the mindset that objective standards to teaching math was the norm.  Course material has always been presented to me as an absolute reality which was pretty standard in all schools.  And as far as the content was concerned, I thought that the material taught to us was the only appropriate material for our age group.
In high school, I did get some newer perspectives, however, on the different ways of teaching mathematics, because students from Asian countries seemed to have had a better mathematical education (at least in terms of the level of difficulty or grade level).  However, I do not know how teachers in Asia differered in their pedagogical approach.
I find this article so interesting because it challenges my view that the way I was taught mathematics was the only way that had been tried up to then.  But in reality, pedagogical approaches to mathematics have not always been monoform, but diverse.  In fact, it is because of this diversity that correct pedagogy has become in some ways a political issue.
When I teach mathematics, I will certainly take a lot from the "new" ways of teaching - but I do not think that my way will be a totally novel way of teaching.  What I hope is simply to combine philosophy and thinking with mathematics, to reach people at many different levels, rather than at just the end-result level which has test-taking as its ultimate aim.

Lesson Plan for Mini-Lessons

Title: Greek Alphabet and Basic words
Date: October 21 2015
Grade Level: N/A
Prescribed Learning Outcomes: To teach students the 24-letter Greek Alphabet
General Purpose: This class will look at how to write out the letters of the Greek Alphabet, how to pronounce them, and a few basic words in Greek.
SWBAT: q Correctly pronounce all the letters in the Greek Alphabet. q Be able to read a few basic words in Greek.
Probing of previous knowledge: Students are not expected to have any formal training in Greek.  However, many of the Greek letters will be familiar to math students who have seen these letters used as variables.
Objective	Time	Activity	Materials
Introduction/hook	1 minute	Tell the students "good day" in Greek! Tell the students what I will be introducing to them.
Alphabet activity	5 minutes	With the students, go through the enter Greek Alphabet (all 24 letters) Write the letters on the board, and have the student write them out, while pronouncing them	·     paper and pencil
Explanation of friction	3 minutes	Write some basic Greek words on the board, and try to have the students read them out correctly. Note how close some Greek words come to English derivations of them!
Conclusion	1 minute	Present a Greek phrase to the students, taken from the inscription above Plato's Academy.  "Let no one who knows not geometry enter here!"  Read it out.
Summative Evaluation: By having all the students write out the 24 Greek letters, I know that all the students have thought about and applied their basic knowledge of Greek.  They can now present Greek letters with more confidence in a math classroom!
References: http://www.ibiblio.org/koine/greek/lessons/alphabet.html https://en.wikipedia.org/wiki/Greek_alphabet

Estimating the volume of a campbell soup water tank

First of all, I want to point out that pictures are deceiving.  Although it looks like the bottom of the bike wheel starts below the lowest point of the water tank, one can see on closer examination that part of the tank is buried in sand (understandably, to keep the tank from rolling away!)  Furthermore, since the top of the water tank is further away than the bike, we have to adjust our perspective slightly so as to take this into account.  Overall, I would estimate that the diameter of the water tank is 2.2 times the height of the bike.
To determine the height of Susan's bike, I measured the height of a medium-frame bike I have at home.  It was 102 cm.
The diameter of the water tank is therefore 102cm x 2.2 = 224cm.
According to an online source (https://sketchup-math.wikispaces.com/Campbell's+Soup), a Campbell soup can has a height of 4 inches and a radius of 1 and 5/16 inches.  Therefore it has a diameter of 2 and 5/8 inches.  Therefore, the height is 4/(2+5/8)=1.52 times the diameter.  Apply this now to the water tank.  The water tank's length (equivalent to a can's "height") will be 1.52x224cm=341cm.
The volume of the tank will therefore be pi*radius^2*height = pi*(224/2)^2*341 = 13450000 cm^3 = 13450000mL = 13 500L = 3550 gallons!

Imaginary letters from former students & reflection

Dear Mr. O'Toole,

This is Timmy.  I was a grade 10 student in your math class 10 years ago!  I was thinking about you recently and wanted to write a little note.  I don't know if you knew this, but after grade 10 I dropped out of the foundations of math to the essentials of math class.  I always found it difficult to keep up in class.  The rate at which you went through the material, though good for others, was always way too fast for me.  And then, there was the homework.  Oh, you assigned so much homework!  I could never get through more than half of it.  My parents could not afford a tutor for me, so I had no choice but to drop foundations math.  Though now, I sort of wished that I could have received some more help, so that universities would have looked better upon my report cards!

Timmy.


Dear Mr. O'Toole,

This is Amanda.  I was part of your grade 12 math class 10 years ago.  I just wanted to write to say how much I enjoyed math in your class, and how satisfying it was to be able to understand the material you gave us.  You always went at a good pace, with clear explanations, so that I was able to keep up with all the school work and even have time to be part of the math club.  You seemed to love math a lot, and communicated it in a passionate way.  Thank you for helping me to do well in class, which prepared me for later in the future.

Amanda

Reflection:
Perhaps my greatest struggle in teaching will have to do with the different levels of ability in math that I will encounter among students.  Some students will be stronger, others not so strong.  I think that I am going to have to find a way to communicate the material to all, while at the same time adapting it to certain learners through ways of teaching and through other strategies.  It will take experience, I think, to be very in-tune to the students' needs, and to learn different approaches which will help all the students in my class.  The issue about homework is a big one.  I will somehow have to try to keep my students accountable to the work that I give them.
This activity helped me to think from the perspective of the student, who does not always perceive things in the same way as the teacher does.  The teacher needs to be very understanding of every individual person.

Reflection on math art presentation - hyperboloids

I have never made a math-art project before.  I didn't know what to expect.  For me, mathematics throughout secondary school and university was about abstract concepts and theories.  So one can well imagine the benefit I gained from actually doing a hands-on math project.

First of all, there was the social dimension of the project.  Working in groups was fun, and was conducive to making new friends.  I think that in high school, doing a project like this (especially in earlier years when kids still don't know one another too well) would be very useful in building bonds and relationships between students, helping them to make friends.  Mathematics doesn't just have to be about formulae, it can also be fun.

As far as the hyperboloid structure is concerned, I learned a lot about mathematics.  I learned that a hyperboloid can be made out of straight lines, that these lines are parallel to one another, and that a line only intersects any other line (if they are not parallel) in one place.  I learned that one can take a two-dimensional lattice of skewers, and shape it in three-dimensions to form a hyperboloid.  The concept of a hyperboloid is therefore more comprehensible and is not outside my ability to understand.

I think that such a project can make certain areas of mathematics more comprehensible for students who have never though of mathematics outside the realm of paper and pencil.

Math That Matters reflection

This article was, I thought, very interesting.  It approaches mathematics from a social justice perspective.  This idea is not unlike ideas presented in other classes, whereby teachers might introduce social justice in all school classes.

My opinion regarding this matter might be somewhat controversial.  Following discussions I've had with students from other classes, I feel that teachers should not become political ideologues.  Although social justice certainly has a place in the curriculum, and ought to form the perspective of the teacher and of the students, I am wary about actually regimenting it so that it is the basis of the educational approach.  That is not to say, however, that I disagree with applying any of David Stocker's ideas in the classroom.  What I am hesitant to agree with, however, is when David Stocker speaks about applying these ideas in the classroom even to the disagreement of the parents.  I am a supporter of the idea that parents are the primary educators of their children, and I think that, despite the good intentions of teachers, a political approach to education would ultimately be too narrowing.  Children are more than political entities whose potential depends on how much they might change or influence society in the future.  As a teacher, I will be dedicated to forming individual students, not for the sake of political ideals, but for their sake.